Dual representations for systemic risk measures

Abstract

The financial crisis showed the importance of measuring, allocating and regulating systemic risk. Recently, the systemic risk measures that can be decomposed into an aggregation function and a scalar measure of risk, received a lot of attention. In this framework, capital allocations are added after aggregation and can represent bailout costs. More recently, a framework has been introduced, where institutions are supplied with capital allocations before aggregation. This yields an interpretation that is particularly useful for regulatory purposes. In each framework, the set of all feasible capital allocations leads to a multivariate risk measure. In this paper, we present dual representations for scalar systemic risk measures as well as for the corresponding multivariate risk measures concerning capital allocations. Our results cover both frameworks: aggregating after allocating and allocating after aggregation. As examples, we consider the aggregation mechanisms of the Eisenberg-Noe model as well as those of the resource allocation and network flow models. Finally, we illustrate how the dual representations developed here can be useful for computational purposes.

Multi-utility representations of incomplete preferences induced by set-valued risk measures

“…The result provides an equivalent formulation of the dual representation of sensitive systemic set-valued risk measures established in Ararat and Rudloff [1].…”

“…This note is concerned with the numerical representation of preference relations induced by a special class of set-valued maps. Recall that a preference (relation) over the elements of a set L is a reflexive and transitive binary relation on L. A preference is said to be complete if any two elements x, y ∈ L are comparable in the sense that B C. Munari cosimo.munari@bf.uzh.ch 1 Center for Finance and Insurance and Swiss Finance Institute, Department of Banking and Finance, University of Zurich, Plattenstrasse 14, 8032 Zurich, Switzerland it is always possible to determine whether x is preferred to y or vice versa. Following the terminology of Dubra et al [14], a family U of maps u : L → [−∞, ∞] is a multi-utility representation of a preference if for all x, y ∈ L, we have…”

“…Here, we used the standard double arrow notation ⇒ to denote a map taking values in the power set of M. The map R can be seen as a generalisation of the set-valued risk measures studied by Jouini et al [32], Kulikov [35], Hamel and Heyde [28], Hamel et al [30] and Molchanov and Cascos [37] in the context of markets with transaction costs; by Haier et al [27] in the context of intragroup transfers; by Feinstein et al [22], Armenti et al [3] and Ararat and Rudloff [1] in the context of systemic risk. We refer to these contributions for a discussion about the financial interpretation of set-valued risk measures in the respective fields of application and to Sect.…”

Optimal Network Compression

“…These functions are decomposed as ρ • Λ for a risk measure ρ and an aggregation function Λ. Such functions were first introduced in [15,38] and are detailed below; these mappings also coincide with the "insensitive systemic risk measures" of [14,28]. In order to present this setting, and for the remainder of this paper, we fix some probability space (Ω, F, P).…”

Set-Valued Dynamic Risk Measures for Bounded Discrete-Time Processes

“…Set-valued risk measures were introduced so as to consider risks in markets with frictions. Such risk measures have, more recently, been utilized for quantifying systemic risk in [16,2]. In [7,8], set-valued risk measures for processes were introduced.…”